Definitely replace that old RG59. RG 6 is much better as it has less signal loss and the old RG 59 no doubt has some oxidation. You'll get a much better signal for little money. The new coax is cheap too so you may as well use quad shield.
I've put up coax a few times the last several years and making a new fm antenna run I noticed oxidation on 2 year old coax even though it had been very well sealed. Fortunately it was a longer run and I'd planned on just replacing the whole run anyway but had been really surprised by the oxidation on the shield.
Home Depot sells it by the foot. Seems like it's about 30 or 40 cents a foot for the quad stuff. If it's going to get any weather there's F connectors for that environment too. I'd use it anyway as they don't cost much.
The run length can be significant but it sounds like you wouldn't have much. If you can crimp on your own connectors you can make everything the length you need. Be sure to get connectors for RG 6 and not RG 59 as they won't fit and if you don't read the package like one guy I know

you'll have to go get some more.